Exportation[1][2][3][4] is a valid rule of replacement in
propositional logic. The rule allows conditional statements having
conjunctive antecedents to be replaced by statements having
conditional consequents and vice versa in logical proofs. It is the
rule that:

(
(
P
∧
Q
)
→
R
)
⇔
(
P
→
(
Q
→
R
)
)

displaystyle ((Pland Q)to R)Leftrightarrow (Pto (Qto R))

Where "

⇔

displaystyle Leftrightarrow

" is a metalogical symbol representing "can be replaced in a proof
with."

Contents

1 Formal notation
2 Natural language

2.1 Truth values
2.2 Example

3 Proof
4 Relation to functions
5 References

Formal notation[edit]
The exportation rule may be written in sequent notation:

(
(
P
∧
Q
)
→
R
)
⊣⊢
(
P
→
(
Q
→
R
)
)

displaystyle ((Pland Q)to R)dashv vdash (Pto (Qto R))

where

⊣⊢

displaystyle dashv vdash

is a metalogical symbol meaning that

(
P
→
(
Q
→
R
)
)

displaystyle (Pto (Qto R))

is a syntactic equivalent of

(
(
P
∧
Q
)
→
R
)

displaystyle ((Pland Q)to R)

in some logical system;
or in rule form:

(
P
∧
Q
)
→
R

P
→
(
Q
→
R
)

displaystyle frac (Pland Q)to R Pto (Qto R)

,

P
→
(
Q
→
R
)

(
P
∧
Q
)
→
R

.

displaystyle frac Pto (Qto R) (Pland Q)to R .

where the rule is that wherever an instance of "

(
P
∧
Q
)
→
R

displaystyle (Pland Q)to R

" appears on a line of a proof, it can be replaced with "

P
→
(
Q
→
R
)

displaystyle Pto (Qto R)

" and vice versa;
or as the statement of a truth-functional tautology or theorem of
propositional logic:

(
(
P
∧
Q
)
→
R
)
↔
(
P
→
(
Q
→
R
)
)

displaystyle ((Pland Q)to R)leftrightarrow (Pto (Qto R))

where

P

displaystyle P

,

Q

displaystyle Q

, and

R

displaystyle R

are propositions expressed in some logical system.
Natural language[edit]
Truth values[edit]
At any time, if P→Q is true, it can be replaced by P→(P∧Q).
One possible case for P→Q is for P to be true and Q to be true; thus
P∧Q is also true, and P→(P∧Q) is true.
Another possible case sets P as false and Q as true. Thus, P∧Q is
false and P→(P∧Q) is false; false→false is true.
The last case occurs when both P and Q are false. Thus, P∧Q is false
and P→(P∧Q) is true.
Example[edit]
It rains and the sun shines implies that there is a rainbow.
Thus, if it rains, then the sun shines implies that there is a
rainbow.
Proof[edit]
The following proof uses Material Implication, double negation, De
Morgan's Laws, the negation of the conditional statement, the
Associative Property of conjunction, the negation of another
conditional statement, and double negation again, in that order to
derive the result.

Proposition
Derivation

P
→
(
Q
→
R
)

displaystyle Prightarrow (Qrightarrow R)

Given

¬
P
∨
(
Q
→
R
)

displaystyle neg Plor (Qrightarrow R)

Material implication

¬
P
∨
¬
¬
(
Q
→
R
)

displaystyle neg Plor neg neg (Qrightarrow R)

double negation

¬
[
P
∧
¬
(
Q
→
R
)
]

displaystyle neg [Pland neg (Qrightarrow R)]

De Morgan's law

¬
[
P
∧
(
Q
∧
¬
R
)
]

displaystyle neg [Pland (Qland neg R)]

Negation of Conditional

¬
[
(
P
∧
Q
)
∧
¬
R
]

displaystyle neg [(Pland Q)land neg R]

Associativity

¬
[
¬
(
(
P
∧
Q
)
→
R
)
]

displaystyle neg [neg ((Pland Q)rightarrow R)]

Negation of Conditional

(
P
∧
Q
)
→
R

displaystyle (Pland Q)rightarrow R

double negation

Relation to functions[edit]
Exportation is associated with
Currying via the Curry–Howard
correspondence.
References[edit]